A spectrograph is an optical instrument used to disperse and sharply focus light in the plane of dispersion, typically the horizontal or tangential plane of the instrument, onto a focal plane array detector. For further clarification, the tangential plane herein refers to the plane parallel to the page. Spectrographs are typically used to investigate specific material properties through light's various interactions with matter. Several examples include, though not limited to, Raman scattering, fluorescence emission/excitation spectroscopy, Rayleigh scattering, etc. . . . Modern commercial spectrographs typically combine one or more curved optical elements, either reflective mirrors or refractive lenses, which collimate light to and focus dispersed light from a dispersive element, such as a diffraction grating or prism. Light consisting of a plurality of dispersed wavelengths is focused onto a focal plane array detector, such as a charge coupled device (CCD) or photo diode array (PDA).
Typical commercial spectrographs employ the Czerny-Turner type optical design or variants thereof. In this design, two mirrors are used with off-axis chief rays with a dispersive element placed near their midpoint to form a ‘W’ shape. More specifically, the two mirrors are located at the bottom apexes of the W and the grating at the top apex. The first mirror, typically a toroid in shape, collimates light from a source point located at the entrance slit of the spectrograph. The source point may be a fiber optic, multiple fiber optics placed at the slit plane, or an image projected from any optical instrument. A dispersive element, usually a diffraction grating, is arranged to receive collimated light from the first mirror and disperse collimated light towards the second mirror. The second mirror, typically spherical in shape, focuses spectrally dispersed images of the source point with residual aberrations onto a focal plane array detector. These residual image aberrations are inherent in typical Czerny-Turner designs and are a defining characteristic of the instrument.
The imaging performance of a Czerny-Turner spectrograph correlates to how well it will resolve dispersed spectral features and the extent to which source points located vertically along the slit plane may be spatially resolved. Spatial resolution along the slit plane is of paramount importance for multi-channel spectroscopy or hyper-spectral imaging techniques. The three primary third order ‘Seidel’ aberrations that limit imaging performance that concern one designing a spectrograph, listed here by their Seidel coefficient, are spherical (SI), coma (SII), and astigmatism (SIII). Of these three aberrations, coma and astigmatism are the most critical to the designer because they asymmetrically distort recorded spectral features and affect both dispersive and spatial resolution. Spherical aberration, or SI, is less concerning because it symmetrically broadens line profiles resulting in diminished peak intensity in a spectral feature.
Uncorrected SI in a typical Czerny-Turner spectrograph is observed as a diffuse symmetric blur about the image of a source point and is known to increase in severity as 1/(f/#)3. As used herein, f/# or ‘f number’, refers to the ratio of a mirror or lens's effective focal length to the diameter of its entrance pupil. The f/# of a mirror also correlates to its light collecting power as 1/(f/#)2. Therefore, the smaller the f/# of a spectrograph, the faster it will gather light and the more prone it becomes to suffering from debilitating image aberrations.
It is known historically from the Rayleigh Criterion that the wavefront aberration, WI, caused by SI alone should be made less than λ/4 to insure diffraction limited performance in an optical system. As used herein, WI is the wavefront aberration produced by SI and λ a particular wavelength of light. For large aperture low f/# mirrors, for example, mirrors having an f/# lower than f/5 with diameters greater than 32 mm operating at a design wavelength λ of 500 nm, will suffer noticeable WI and correction should be implemented into the optical design of the spectrograph.
Mathematically, the Seidel coefficient SI for a spherical mirror is listed as equation 1 where ‘y’ is the radial distance measured from the mirror apex out to the clear aperture edge and ‘R’ the radius of curvature. All subscripts refer to the respective mirror in question and the sum over all like Seidel coefficients gives the total respective aberration in the optical train comprising the spectrograph. The wavefront aberration associated with SI, labeled WI, is given by equation 2 where ym is the mirror's maximum clear aperture half-width. Because SI and WI respectively increase as the 4'th power in mirror half-width, WI rapidly becomes problematic for large aperture, low f/# optics.
                                          (            SI            )                    i                =                  2          ⁢                                    y              i              4                                      R              i              2                                                          (        1        )                                                      (            WI            )                    i                =                              1            8                    ⁢                                    (                              y                                  y                  m                                            )                        4                    ⁢                                    (              SI              )                        i                                              (        2        )            Uncorrected SII is observed as the asymmetric broadening of the image of a source point primarily in the tangential or dispersion plane of the spectrograph. SII is caused by chief rays reflecting from a mirror rotated about its optical axis. In the case of the Czerny-Turner spectrograph, mirrors are rotated about the sagittal or vertical axis which predominantly adds positive or negative tangential SII into the image. Sagittal SII is present, however, to a much lesser extent and is of little concern. Mathematically, the SII coefficient for a spherical mirror is represented by equation 3 where si is the distance along the principal ray traced from the mirror's vertex to the center of the system stop, i.e. the grating, and u the principal ray angle or the off-axis angle on the mirror.
                                          (            SII            )                    i                =                              -            2                    ⁢                                    (                                                y                  i                                                  R                  i                                            )                        3                    ⁢                      (                                          R                i                            -                              s                i                                      )                    ⁢          sin          ⁢                                          ⁢                      u            i                                              (        3        )            Uncorrected SIII is observed as the asymmetric broadening of the image of a source point in the sagittal or vertical plane when a detector is positioned for maximum resolution or tightest sagittal focus. SIII is the result of the tangential and sagittal focal planes for a concave mirror departing longitudinally from one another when arranged to image off-axis source points. SIII for all non-axial image points, or field points, is observed to increase rapidly in the typical Czerny-Turner spectrograph with increasing tangential image distance from the focal plane center. As used herein, the term ‘field’ refers to any image point or aberration of an image point formed a measurable distance from the center of the focal plane. The fluence in recorded spectral images then decreases for all field points because the image of the source point becomes vertically elongated covering more image sensing pixels. Mathematically, the SIII coefficient for a spherical mirror is defined as equation 4.
                                          (            SIII            )                    i                =                                            (                                                y                  i                                                  R                  i                                            )                        2                    ⁢                                    2                              R                i                                      ⁡                          [                                                                    R                    i                                    ⁡                                      (                                                                  R                        i                                            -                                              2                        ⁢                                                  s                          i                                                                                      )                                                  +                                  s                  i                  2                                            ]                                ⁢          sin          ⁢                                          ⁢                      u            i            2                                              (        4        )            
In the typical Czerny-Turner spectrograph, methods for correcting for axial SII and SIII have been realized whereas correction for SI is typically absent and designers have historically followed the Rayleigh Criterion as a rough design guide. However, this rule warns against the use of low f/#, or fast optics, having long focal lengths. Because the dispersive resolution in a spectrograph is proportional to the focal length of its focusing mirror, a fast, high resolution instrument, absent of SI is not possible if using a conventional design.
It is known that axial SII can be entirely corrected at one grating angle by correct choice of mirror radii Ri and off-axis angles ui. This is evident from equation 3 for the sign of the off axis angle ui will reverse for the collimating and focusing mirrors in the conventional ‘W’ arrangement. Therefore, a condition can be met where the coma introduced by the first mirror is equal and opposite that of the second. However, the diffraction grating imparts anamorphic magnification into the dispersed beam which compresses or expands the beam and, most importantly, this anamorphic effect changes with grating angle. Therefore, the half-width of the beam illuminating the second mirror is a function of grating angle and so SII can only be corrected for a specific design grating angle or rather, design wavelength range.
SIII is typically corrected for axial image points only, that is, it only tends to zero at the center of the focal plane and field SIII is left uncorrected. It is known that axial SIII correction can be accomplished in several ways. The most common method for correcting axial SIII is the use of a toroidal collimating mirror which has a shorter radius of curvature in the sagittal plane than the tangential plane. The choice of optimum sagittal radius is determined by considering the total astigmatic focal shift imparted by the two concave mirrors used at their respective off-axis angles ui. The total astigmatic focal shift for two concave mirrors each having one infinite conjugate plane and arranged in such a way as to image a source point located a distance ft1 from the first mirror is given as equation 5a. Sagittal and tangential focal lengths, fs and ft, are related to a mirror's sagittal and tangential radius of curvature Rs and Rt, if toroidal, and are given by equations 5b and 5c. Numerical and index ‘i’ subscripts in equations 5a-5c refer to the first ‘collimating’ mirror and second ‘focusing’ mirror. Note that for a spherical mirror Rs is equal to Rt, however, fs and ft are not equal due to a non-zero off-axis angle u. The sagittal radius on the collimating mirror Rs1 may be determined according to equation 5a for zero astigmatic focal shift. That is, ΔfSIII=0. This method will remove axial astigmatism from the final image.
                              Δ          ⁢                                          ⁢                      f            SIII                          =                              (                                          f                                  s                  ⁢                                                                          ⁢                  1                                            -                              f                                  t                  ⁢                                                                          ⁢                  1                                                      )                    +                      (                                          f                                  s                  ⁢                                                                          ⁢                  2                                            -                              f                                  t                  ⁢                                                                          ⁢                  2                                                      )                                              (                  5          ⁢          a                )                                          f                      s            i                          =                              R                          s              i                                            2            ⁢                                                  ⁢            cos            ⁢                                                  ⁢                          u              i                                                          (                  5          ⁢          b                )                                          f                      t            i                          =                                            R                              t                i                                      2                    ⁢          cos          ⁢                                          ⁢                      u            i                                              (                  5          ⁢          c                )            
In place of a toroidal collimating mirror, the grating, having uniform groove spacing, may itself be toroidal in shape so as to provide the necessary condition for axial SIII correction per equation 5a. In this configuration, the toroidal grating takes the place of the collimating mirror and provides axial SIII correction at one wavelength or more precisely at one grating angle. As the grating is rotated from the ideal angle, so as to change the observed wavelength range spanned by the focal plane array detector, correction for axial SIII will suffer.
A third method for correcting axial SIII includes using an aberration corrected holographic grating having variable line spacing. Such gratings can completely correct for axial SIII at one wavelength and moderately suppress axial SIII at other wavelengths. (U.S. Pat. No. 3,628,849) All references cited herein are incorporated by reference as if fully set forth herein.
Uncorrected field SIII in a spectrograph is highly detrimental when spatial resolution for source points located vertically along the entrance slit is desired. For example, if multiple fiber optic sources from a linear fiber bundle are placed at the slit plane, uncorrected field SIII will result in dispersed light from adjacent fiber optic sources to overlap or ‘cross-talk’ at the edges of the focal plane. This ultimately reduces the number of fiber optic sources or discrete optical channels an imaging spectrograph can accommodate before cross-talk occurs. Additionally, in the case where an image projected from a microscope or any other image forming instrument is incident at the entrance slit plane of the spectrograph, uncorrected SIII will result in the inability to resolve spatial image information for field points in the sagittal plane.
It is therefore desirable to provide a high resolution imaging spectrograph that operates at low f/# and which provides anastigmatic imaging over the entire field of a flat focal plane array detector at its design wavelength and remains nearly anastigmatic for wavelengths departing from its design wavelength.